Squeeze-film flows, with application to the human knee joint

The squeeze-film flow of a thin layer of Newtonian fluid filling the gap between either a flat or a curved rigid impermeable solid coated with a thin rigid or elastic layer and moving under a prescribed constant load towards a thin rigid porous bed of uniform thickness coating a stationary flat impermeable solid is considered. Three related problems are considered separately, namely the case of a flat surface coated with a rigid layer, the case of a curved surface coated with a rigid layer and the general case of a curved surface coated with an elastic layer. Unlike the case of an impermeable bed, in which an infinite time is required for the two surfaces to come into contact, for a porous bed contact occurs in a finite time. Lubrication approximations are used and numerical and (where possible) analytical solutions for the fluid layer thickness, the fluid pressure and the contact time are obtained and analysed. Furthermore, in the rigid cases the fluid particle paths are calculated, and the penetration depths of fluid particles into the porous bed are determined. In particular, the behaviour in the asymptotic limit of small permeability, in which the contact time is large but finite, is investigated. Finally, the results are interpreted in the context of lubrication in the human knee joint, and some conclusions are drawn about the contact time of the cartilage-coated femoral condyles and tibial plateau and the penetration of nutrients into the cartilage.

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